True Infinitesimal Differential Geometry Mikhail G. Katz∗ Vladimir
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Differential Calculus and by Era Integral Calculus, Which Are Related by in Early Cultures in Classical Antiquity the Fundamental Theorem of Calculus
History of calculus - Wikipedia, the free encyclopedia 1/1/10 5:02 PM History of calculus From Wikipedia, the free encyclopedia History of science This is a sub-article to Calculus and History of mathematics. History of Calculus is part of the history of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The subject, known Background historically as infinitesimal calculus, Theories/sociology constitutes a major part of modern Historiography mathematics education. It has two major Pseudoscience branches, differential calculus and By era integral calculus, which are related by In early cultures in Classical Antiquity the fundamental theorem of calculus. In the Middle Ages Calculus is the study of change, in the In the Renaissance same way that geometry is the study of Scientific Revolution shape and algebra is the study of By topic operations and their application to Natural sciences solving equations. A course in calculus Astronomy is a gateway to other, more advanced Biology courses in mathematics devoted to the Botany study of functions and limits, broadly Chemistry Ecology called mathematical analysis. Calculus Geography has widespread applications in science, Geology economics, and engineering and can Paleontology solve many problems for which algebra Physics alone is insufficient. Mathematics Algebra Calculus Combinatorics Contents Geometry Logic Statistics 1 Development of calculus Trigonometry 1.1 Integral calculus Social sciences 1.2 Differential calculus Anthropology 1.3 Mathematical analysis -
Calculus? Can You Think of a Problem Calculus Could Be Used to Solve?
Mathematics Before you read Discuss these questions with your partner. What do you know about calculus? Can you think of a problem calculus could be used to solve? В A Vocabulary Complete the definitions below with words from the box. slope approximation embrace acceleration diverse indispensable sphere cube rectangle 1 If something is a(n) it В Reading 1 isn't exact. 2 An increase in speed is called Calculus 3 If something is you can't Calculus is the branch of mathematics that deals manage without it. with the rates of change of quantities as well as the length, area and volume of objects. It grew 4 If you an idea, you accept it. out of geometry and algebra. There are two 5 A is a three-dimensional, divisions of calculus - differential calculus and square shape. integral calculus. Differential calculus is the form 6 Something which is is concerned with the rate of change of quantities. different or of many kinds. This can be illustrated by slopes of curves. Integral calculus is used to study length, area 7 If you place two squares side by side, you and volume. form a(n) The earliest examples of a form of calculus date 8 A is a three-dimensional back to the ancient Greeks, with Eudoxus surface, all the points of which are the same developing a mathematical method to work out distance from a fixed point. area and volume. Other important contributions 9 A is also known as a fall. were made by the famous scientist and mathematician, Archimedes. In India, over the 98 Macmillan Guide to Science Unit 21 Mathematics 4 course of many years - from 500 AD to the 14th Pronunciation guide century - calculus was studied by a number of mathematicians. -
Cauchy, Infinitesimals and Ghosts of Departed Quantifiers 3
CAUCHY, INFINITESIMALS AND GHOSTS OF DEPARTED QUANTIFIERS JACQUES BAIR, PIOTR BLASZCZYK, ROBERT ELY, VALERIE´ HENRY, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, TARAS KUDRYK, SEMEN S. KUTATELADZE, THOMAS MCGAFFEY, THOMAS MORMANN, DAVID M. SCHAPS, AND DAVID SHERRY Abstract. Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson’s frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz’s distinction be- tween assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robin- son’s framework, while Leibniz’s law of homogeneity with the im- plied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide paral- lel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz’s infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Eu- ler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infi- nite exponent. Such procedures have immediate hyperfinite ana- logues in Robinson’s framework, while in a Weierstrassian frame- work they can only be reinterpreted by means of paraphrases de- parting significantly from Euler’s own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson’s framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of de- arXiv:1712.00226v1 [math.HO] 1 Dec 2017 parted quantifiers in his work. -
Historical Notes on Calculus
Historical notes on calculus Dr. Vladimir Dotsenko Dr. Vladimir Dotsenko Historical notes on calculus 1/9 Descartes: Describing geometric figures by algebraic formulas 1637: Ren´eDescartes publishes “La G´eom´etrie”, that is “Geometry”, a book which did not itself address calculus, but however changed once and forever the way we relate geometric shapes to algebraic equations. Later works of creators of calculus definitely relied on Descartes’ methodology and revolutionary system of notation in the most fundamental way. Ren´eDescartes (1596–1660), courtesy of Wikipedia Dr. Vladimir Dotsenko Historical notes on calculus 2/9 Fermat: “Pre-calculus” 1638: In a letter to Mersenne, Pierre de Fermat explains what later becomes the key point of his works “Methodus ad disquirendam maximam et minima” and “De tangentibus linearum curvarum”, that is “Method of discovery of maximums and minimums” and “Tangents of curved lines” (published posthumously in 1679). This is not differential calculus per se, but something equivalent. Pierre de Fermat (1601–1665), courtesy of Wikipedia Dr. Vladimir Dotsenko Historical notes on calculus 3/9 Pascal and Huygens: Implicit calculus In 1650s, slightly younger scientists like Blaise Pascal and Christiaan Huygens also used methods similar to those of Fermat to study maximal and minimal values of functions. They did not talk about anything like limits, but in fact were doing exactly the things we do in modern calculus for purposes of geometry and optics. Blaise Pascal (1623–1662), courtesy of Christiaan Huygens (1629–1695), Wikipedia courtesy of Wikipedia Dr. Vladimir Dotsenko Historical notes on calculus 4/9 Barrow: Tangents vs areas 1669-70: Isaac Barrow publishes “Lectiones Opticae” and “Lectiones Geometricae” (“Lectures on Optics” and “Lectures on Geometry”), based on his lectures in Cambridge. -
Evolution of Mathematics: a Brief Sketch
Open Access Journal of Mathematical and Theoretical Physics Mini Review Open Access Evolution of mathematics: a brief sketch Ancient beginnings: numbers and figures Volume 1 Issue 6 - 2018 It is no exaggeration that Mathematics is ubiquitously Sujit K Bose present in our everyday life, be it in our school, play ground, SN Bose National Center for Basic Sciences, Kolkata mobile number and so on. But was that the case in prehistoric times, before the dawn of civilization? Necessity is the mother Correspondence: Sujit K Bose, Professor of Mathematics (retired), SN Bose National Center for Basic Sciences, Kolkata, of invenp tion or discovery and evolution in this case. So India, Email mathematical concepts must have been seen through or created by those who could, and use that to derive benefit from the Received: October 12, 2018 | Published: December 18, 2018 discovery. The creative process of discovery and later putting that to use must have been arduous and slow. I try to look back from my limited Indian perspective, and reflect up on what might have been the course taken up by mathematics during 10, 11, 12, 13, etc., giving place value to each digit. The barrier the long journey, since ancient times. of writing very large numbers was thus broken. For instance, the numbers mentioned in Yajur Veda could easily be represented A very early method necessitated for counting of objects as Dasha 10, Shata (102), Sahsra (103), Ayuta (104), Niuta (enumeration) was the Tally Marks used in the late Stone Age. In (105), Prayuta (106), ArA bud (107), Nyarbud (108), Samudra some parts of Europe, Africa and Australia the system consisted (109), Madhya (1010), Anta (1011) and Parartha (1012). -
Fermat, Leibniz, Euler, and the Gang: the True History of the Concepts Of
FERMAT, LEIBNIZ, EULER, AND THE GANG: THE TRUE HISTORY OF THE CONCEPTS OF LIMIT AND SHADOW TIZIANA BASCELLI, EMANUELE BOTTAZZI, FREDERIK HERZBERG, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, TAHL NOWIK, DAVID SHERRY, AND STEVEN SHNIDER Abstract. Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding “negligible” terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern the- ories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions. Contents 1. Introduction 2 2. Methodological remarks 4 2.1. A-track and B-track 5 2.2. Formal epistemology: Easwaran on hyperreals 6 2.3. Zermelo–Fraenkel axioms and the Feferman–Levy model 8 2.4. Skolem integers and Robinson integers 9 2.5. Williamson, complexity, and other arguments 10 2.6. Infinity and infinitesimal: let both pretty severely alone 13 3. Fermat’s adequality 13 3.1. Summary of Fermat’s algorithm 14 arXiv:1407.0233v1 [math.HO] 1 Jul 2014 3.2. Tangent line and convexity of parabola 15 3.3. Fermat, Galileo, and Wallis 17 4. Leibniz’s Transcendental law of homogeneity 18 4.1. When are quantities equal? 19 4.2. Product rule 20 5. Euler’s Principle of Cancellation 20 6. What did Cauchy mean by “limit”? 22 6.1. Cauchy on Leibniz 23 6.2. Cauchy on continuity 23 7. Modern formalisations: a case study 25 8. A combinatorial approach to decreasing rearrangements 26 9. -
Pure Mathematics
good mathematics—to be without applications, Hardy ab- solved mathematics, and thus the mathematical community, from being an accomplice of those who waged wars and thrived on social injustice. The problem with this view is simply that it is not true. Mathematicians live in the real world, and their mathematics interacts with the real world in one way or another. I don’t want to say that there is no difference between pure and ap- plied math. Someone who uses mathematics to maximize Why I Don’t Like “Pure the time an airline’s fleet is actually in the air (thus making money) and not on the ground (thus costing money) is doing Mathematics” applied math, whereas someone who proves theorems on the Hochschild cohomology of Banach algebras (I do that, for in- † stance) is doing pure math. In general, pure mathematics has AVolker Runde A no immediate impact on the real world (and most of it prob- ably never will), but once we omit the adjective immediate, I am a pure mathematician, and I enjoy being one. I just the distinction begins to blur. don’t like the adjective “pure” in “pure mathematics.” Is mathematics that has applications somehow “impure”? The English mathematician Godfrey Harold Hardy thought so. In his book A Mathematician’s Apology, he writes: A science is said to be useful of its development tends to accentuate the existing inequalities in the distri- bution of wealth, or more directly promotes the de- struction of human life. His criterion for good mathematics was an entirely aesthetic one: The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colours or the words must fit together in a harmo- nious way. -
Connes on the Role of Hyperreals in Mathematics
Found Sci DOI 10.1007/s10699-012-9316-5 Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics Vladimir Kanovei · Mikhail G. Katz · Thomas Mormann © Springer Science+Business Media Dordrecht 2012 Abstract We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in func- tional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “vir- tual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnera- ble to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes’ magnum opus) V. -
Lecture 25: Ultraproducts
LECTURE 25: ULTRAPRODUCTS CALEB STANFORD First, recall that given a collection of sets and an ultrafilter on the index set, we formed an ultraproduct of those sets. It is important to think of the ultraproduct as a set-theoretic construction rather than a model- theoretic construction, in the sense that it is a product of sets rather than a product of structures. I.e., if Xi Q are sets for i = 1; 2; 3;:::, then Xi=U is another set. The set we use does not depend on what constant, function, and relation symbols may exist and have interpretations in Xi. (There are of course profound model-theoretic consequences of this, but the underlying construction is a way of turning a collection of sets into a new set, and doesn't make use of any notions from model theory!) We are interested in the particular case where the index set is N and where there is a set X such that Q Xi = X for all i. Then Xi=U is written XN=U, and is called the ultrapower of X by U. From now on, we will consider the ultrafilter to be a fixed nonprincipal ultrafilter, and will just consider the ultrapower of X to be the ultrapower by this fixed ultrafilter. It doesn't matter which one we pick, in the sense that none of our results will require anything from U beyond its nonprincipality. The ultrapower has two important properties. The first of these is the Transfer Principle. The second is @0-saturation. 1. The Transfer Principle Let L be a language, X a set, and XL an L-structure on X. -
Communications Programming Concepts
AIX Version 7.1 Communications Programming Concepts IBM Note Before using this information and the product it supports, read the information in “Notices” on page 323 . This edition applies to AIX Version 7.1 and to all subsequent releases and modifications until otherwise indicated in new editions. © Copyright International Business Machines Corporation 2010, 2014. US Government Users Restricted Rights – Use, duplication or disclosure restricted by GSA ADP Schedule Contract with IBM Corp. Contents About this document............................................................................................vii How to use this document..........................................................................................................................vii Highlighting.................................................................................................................................................vii Case-sensitivity in AIX................................................................................................................................vii ISO 9000.....................................................................................................................................................vii Communication Programming Concepts................................................................. 1 Data Link Control..........................................................................................................................................1 Generic Data Link Control Environment Overview............................................................................... -
The Hyperreals
THE HYPERREALS LARRY SUSANKA Abstract. In this article we define the hyperreal numbers, an ordered field containing the real numbers as well as infinitesimal numbers. These infinites- imals have magnitude smaller than that of any nonzero real number and have intuitively appealing properties, harkening back to the thoughts of the inven- tors of analysis. We use the ultrafilter construction of the hyperreal numbers which employs common properties of sets, rather than the original approach (see A. Robinson Non-Standard Analysis [5]) which used model theory. A few of the properties of the hyperreals are explored and proofs of some results from real topology and calculus are created using hyperreal arithmetic in place of the standard limit techniques. Contents The Hyperreal Numbers 1. Historical Remarks and Overview 2 2. The Construction 3 3. Vocabulary 6 4. A Collection of Exercises 7 5. Transfer 10 6. The Rearrangement and Hypertail Lemmas 14 Applications 7. Open, Closed and Boundary For Subsets of R 15 8. The Hyperreal Approach to Real Convergent Sequences 16 9. Series 18 10. More on Limits 20 11. Continuity and Uniform Continuity 22 12. Derivatives 26 13. Results Related to the Mean Value Theorem 28 14. Riemann Integral Preliminaries 32 15. The Infinitesimal Approach to Integration 36 16. An Example of Euler, Revisited 37 References 40 Index 41 Date: June 27, 2018. 1 2 LARRY SUSANKA 1. Historical Remarks and Overview The historical Euclid-derived conception of a line was as an object possessing \the quality of length without breadth" and which satisfies the various axioms of Euclid's geometric structure. -
Fermat's Technique of Finding Areas Under Curves
Fermat’s technique of finding areas under curves Ed Staples Erindale College, ACT <[email protected]> erhaps next time you head towards the fundamental theorem of calculus Pin your classroom, you may wish to consider Fermat’s technique of finding expressions for areas under curves, beautifully outlined in Boyer’s History of Mathematics. Pierre de Fermat (1601–1665) developed some impor- tant results in the journey toward the discovery of the calculus. One of these concerned finding areas under simple polynomial curves. It is an amazing fact that, despite deriving these results, he failed to document any connection between tangents and areas. The method is essentially an exercise in geometric series, and I think it is instructive and perhaps refreshing to introduce a little bit of variation from the standard introductions. The method is entirely accessible to senior students, and makes an interesting connection between the anti derivative result and the factorisation of (1 – rn+1). Fermat considered that the area between x = 0 and x = a below the curve y = xn and above the x-axis, to be approximated by circumscribed rectangles where the width of each rectangle formed a geometric sequence. The divi- sions into which the interval is divided, from the right hand side are made at x = a, x = ar, x = ar2, x = ar3 etc., where a and r are constants, with 0 < r < 1 as shown in the diagram. A u s t r a l i a n S e n i o r M a t h e m a t i c s J o u r n a l 1 8 ( 1 ) 53 s e 2 l p If we first consider the curve y = x , the area A contained within the rectan- a t S gles is given by: A = (a – ar)a2 + (ar – ar2)(ar)2 + (ar2 – ar3)(ar2)2 + … This is a geometric series with ratio r3 and first term (1 – r)a3.